| This dissertation addresses the space and time decay of the energy of strong solutions of the homogeneous Navier-Stokes Equations with a well localized initial datum. We use the vorticity of the solution as an auxiliary tool to establish the estimates needed for our results.;First, we consider the Navier-Stokes system and assumptions on the decay of the energy of the solutions and the energy of the weighted solutions in order to prove optimal rates for the weighted higher order derivatives for a extended range of exponents of the weight function. In order to prove this result, we use an interpolation type estimate and modify the weight function for a suitable one, and use certain pressure estimates. These estimates are obtained using singular integrals.;Finally, we prove that the weighted decay of the energy of a solution can be derived from the decay of the energy, assuming that the initial datum is well localized. We extend the range of the exponent of the weight using Fractional Sobolev spaces and some results about singular integrals, using a product rule type result. |
